ОПТИКА И СПЕКТРОСКОПИЯ, 2011, том 111, № 5, с. 770- 779
ФУНКЦИОНАЛЬНЫЕ МАТЕРИАЛЫ ДЛЯ КВАНТОВЫХ ТЕХНОЛОГИЙ
УДК 530.18:535
ON THE ROLE OF INTERBAND SURFACE PLASMONS IN CARBON NANOTUBES1
© 2011 г. I. V. Bondarev*, L. M. Woods*, and A. Popescu**
* Physics Department, North Carolina Central University, Durham, NC 27707, USA **Physics Department, University of South Florida, Tampa, FL 33620, USA e-mail: ibondarev@nccu.edu Received April 18, 2011
Abstract—We review the properties of collective surface excitations — excitons and interband plasmons — in single-walled and double-walled carbon nanotubes. We show that an electrostatic field applied perpendicular to the nanotube axis can control the exciton-plasmon coupling in individual small-diameter (si nm) singlewalled nanotubes, both in the linear excitation regime and in the non-linear excitation regime with the pho-toinduced biexcitonic states formation. For double-walled carbon nanotubes, we report a profound effect of interband surface plasmons on the inter-tube Casimir force at tube separations similar to their equilibrium distances. Strong overlapping plasmon resonances from both tubes warrant their stronger attraction. Nano-tube chiralities possessing such collective excitation features will result in forming the most favorable inner-outer tube combination in double-walled carbon nanotubes. These findings pave the way for the development of new generation of tunable optoelectronic and nano-electromechanical device applications with carbon nanotubes.
INTRODUCTION
Single wall carbon nanotubes (CNs) are quasi-one-dimensional (1D) cylindrical wires made of graphene sheets rolled-up into cylinders of ~1—10 nm in diameter and ~1 ^m up to ~1 cm in length [1, 2]. They combine advantages such as electrical conductivity, chemical stability, high surface area, and unique optoelectronic properties that make them excellent potential candidates for a variety of applications, including efficient solar energy conversion [3], energy storage [4], optical nanobiosensorics [5]. CNs are shown to be very useful as miniaturized electromechanical and chemical devices [6], scanning probe devices [7, 8], and nanomaterials for macroscopic composites [9, 10]. The area of their potential applications was recently expanded towards nanophotonics [11—15], after controllable single-atom incapsulation into singlewalled CNs [16] and controllable exciton emission from pristine single-walled CNs [14] were demonstrated experimentally.
We have reported interactions between excitonic states and surface electromagnetic (EM) fluctuations resulting in the exciton-surface-plasmon coupling in individual small-diameter (<1 nm) semiconducting single-walled CNs [17]. The coupling is a consequence of the nanotube quasi-one-dimensionality that results in the exciton transition dipole moment matrix element and the quasi-momentum vector directed predominantly along the CN axis (the longitu-
1 Доклады XIII международной конференции по квантовой оптике и квантовой информатике (28 мая — 1 июня 2010 г., Киев, Украина.
dinal exciton). This prevents the exciton from the electric dipole coupling to the transversely polarized surface EM modes of the nanotube as they propagate predominantly along the CN axis with their electric vectors orthogonal to the propagation direction. The longitudinally polarized surface EM modes are generated by the electronic Coulomb potential (see, e.g., Ref. [18]), and therefore represent the CN surface plasmon excitations. These have their electric vectors directed along the propagation direction. They do couple to the longitudinal excitons on the CN surface (see Ref. [19] for the complete theory of the phenomenon). Such modes were observed in Ref. [20] to occur in the same energy range of ~1 eV where the exciton excitation energies are located in small-diameter (<1 nm) semiconducting CNs [21]. They are the weakly-dispersive interband plasmon modes [22] similar to the intersubband plasmon modes in quantum wells [23].
The formation of strongly coupled, mixed surface exciton-plasmon excitations is possible when the exciton total energy is in resonance with the energy of an interband surface plasmon mode [13, 17]. The in-depth analysis we performed recently (Ref. [19]) has shown that an electrostatic field applied perpendicular to the CN axis (the quantum confined Stark effect) can affect the exciton and plasmon energies in the way allowing one to control the exciton-plasmon coupling and exciton absorption/emission, accordingly. The optical response of small-diameter CNs exhibits a significant absorption line (Rabi) splitting ~0.1 eV under strong exciton-surface-plasmon coupling as the exci-ton energy is tuned to the nearest interband plasmon
resonance [13, 19]. This is interesting since the strong exciton-plasmon coupling regime occurs here in an individual quasi-lD nanostructure, a semiconducting carbon nanotube, as opposed to artificially fabricated metal-semiconductor nanostructures, such as dye molecules in organic polymers on metallic films [24], semiconductor quantum dots coupled to metallic nanoparticles [25], or nanowires [26], where metal carries the plasmon while semiconductor carries the exciton.
Here, after a brief review of our previous results, we extend our studies to the interactions of biexcitons (observed recently in single-walled CNs by the femtosecond transient absorption spectroscopy technique [27]) with the interband surface plasmon modes. The non-linear absorption lineshapes have been calculated for the (11, 0) CN as an example of a typical small-diameter semiconducting nanotube. They show the characteristic asymmetric splitting behavior as the ex-citon excitation energy is tuned to the nearest interband plasmon resonance of the nanotube. These results, along with those reported earlier for the linear excitonic absorption, should open up paths to new tunable optoelectronic device applications of small-diameter semiconducting CNs, including the strong-excitation regime with optical non-linearities.
We also consider double-wall CN systems to report on our latest findings demonstrating an important role played by low energy (interband) collective surface plasmon excitations in the inter-tube Casimir interaction in these systems. The Casimir interaction is a paradigm for a force induced by quantum electromagnetic (EM) fluctuations. The fundamental nature of this force has been studied extensively ever since the prediction of the existence of an attraction between neutral metallic mirrors in vacuum [28, 29]. In recent years, the Casimir effect has acquired a much broader impact due to its importance for nanostructured materials and devices. The development and operation of micro- and nano-electromechanical systems are limited due to unwanted effects, such as stiction, friction, and adhesion, originating from the Casimir force [30]. This interaction is also an important component for the stability of nanomaterials. To be able to tailor properties of CN based nanomaterials, one has to deeply understand the physics of inter-tube interactions. This is also important for experimental realization of new effects and devices proposed recently, such as trapping of cold atoms [5, 31] and their entanglement [15] near single-walled CNs, surface profiling [7] and nanolithography applications [8] with doublewall CNs.
We show that at tube separations similar to their equilibrium distances interband surface plasmons have a profound effect on the inter-tube Casimir force. Strong overlapping plasmon resonances from both tubes warrant their stronger attraction. Nanotube chiralities possessing such collective excitation fea-
Fig. 1. Schematic of the two concentric CNs in vacuum. The CN radii are and Rj. The regions between the CN surfaces are denoted as (1), (2), and (3).
tures will result in forming the most favorable inner-outer tube combination in double-wall carbon nano-tubes. This theoretical understanding is important for the development of nano-electromechanical devices with CNs.
EXCITON-PLASMON INTERACTIONS AND BIEXCITONIC NONLINEARITIES IN INDIVIDUAL SINGLE-WALLED NANOTUBES
The cylindrical coordinate system is used with the Z-axis directed along the CN axis (see Fig. 1 for the double-walled CN system). The Schrodinger equation for the electron-hole pair on the CN surface in the presence of the perpendicular electrostatic field is solved by separating out the translational and internal degrees of freedom [19], to result in the two differential equations for the transverse motion of the electron and hole in the electrostatic field, and the differential equation for their longitudinal relative motion (representing the exciton) in the effective (field-dependent) Coulomb potential. The former two determine the band-edge Stark shifts field dependences, while the latter one yields the field dependence of the binding energy of the exciton. Following Ref. [32], the effective Coulomb potential is further approximated by the attractive cusp-type cutoff potential, whereby the ex-citon problem becomes mathematically equivalent to that studied in Ref. [32] for 1D semiconductors in zero electrostatic field. The only difference in our case is that the cutoff parameter in our effective Coulomb potential is field-dependent. The exciton excitation energy, Eexc, is the sum of the (negative) binding energy, Eb, and the band gap, Eg, each given by the solution of the corresponding equation. The field dependence of
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Fig. 2. Calculated perpendicular electrostatic field dependences for (left to right): 1) the first bright exciton binding energies in the (11, 0) and (10, 0) CNs (solid lines — exact solutions, dashed lines — second order perturbation theory in the field), 2) the cusp-type cutoff potential in the (11, 0) CN,
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